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Prestressing can produce longitudinal and transverse bend- Schlaich, J., and Scheef, H. (1982) Concrete Box-Girder
ing and shear forces as well as torque in curved box-girder Bridges, ISBN 3 85748 031 9, International Association for
bridges. Torsion, which can increase flexural stresses, must be Bridge and Structural Engineering, Zurich, Switzerland
considered in determining the required prestressing force.
Prestressing can also be used to enhance torsional and trans- This publication is the outcome of a comprehensive survey
verse bending resistance, although this is often avoided for of concrete box-girder bridges. The publication is divided
economic reasons. into three main parts, "Design," "Structural Analysis," and
"Dimensioning and Detailing." A comprehensive reference
Sennah, K. M., and Kennedy, J. B. (2001) "State-of-the-Art list is included. The publication addresses straight, skew, and
in Curved Box-Girder Bridges," Journal of Bridge Engi- curved bridges.
neering, ASCE, Vol. 6, No. 3, pp. 159167. The "Design" section covers several aspects of curved
bridges. Recommendations are given for when the longitudinal
The objective of this paper was to provide highlights of the bending moments can be determined as for a straight bridge
most important references related to the development of cur- and then combined with the torsional effects without consider-
rent guide specifications for the design of straight and curved ing the coupling of the two effects on each other. Alternative
box-girder bridges. As such, it provided an excellent bibliog- substructure configurations are discussed for curved bridges.
raphy from which to identify other papers that were reviewed The "Structural Analysis" section discusses the mutual in-
in detail. Subjects discussed in this review included (1) different fluence of the longitudinal bending moments and torsional
box-girder bridge configurations, (2) construction issues, moments for horizontally curved bridges. A simple table of
(3) deck design, (4) load distribution, (5) deflection and equations based on classical curved beam theory is presented
camber, (6) cross-bracing requirements, (7) end diaphragms, for several different loading conditions of a curved single-
(8) thermal effects, (9) vibration characteristics, (10) impact span bridge with fixed supports.
factors, (11) seismic response, (12) ultimate load-carrying The "Dimensioning and Detailing" section covers several
capacity, (13) buckling of individual components forming the aspects of curved bridges. Dimensioning and reinforcement
box cross section, (14) fatigue, and (15) curvature limitations of the web for flexural shear, torsion, and regional trans-
provided by the codes for treating a curved bridge as a straight verse bending is addressed, including a rational method of
one. The literature survey presented herein encompasses designing the web reinforcement for combined shear and
(1) the construction phase, (2) load distribution, (3) dynamic regional transverse bending. The influence of horizontal
response, and (4) ultimate load response of box-girder bridges. curvature on the movements at bearings is also discussed.
ASCE Committee on Construction Equipment and Tech- Response of Curved Concrete
niques (1989) "Concrete Bridge Design and Construction Box-Girder Bridges
in the United Kingdom," Journal of Construction Engineer-
ing and Management, Vol. 115, No. 4, pp. 618635. Global Analysis
Most published research seems to be directed toward the
The design and construction of concrete bridges in the global response of box-girder bridges. Several analytical tech-
United Kingdom has changed rapidly during recent decades. niques have been studied. Many of these are relatively com-
Better analytical methods, increased mechanization, and plex, but many others are suitable for production design
better planning in the construction of these bridges have practice. Our current belief is that a properly applied grillage
brought this about. However these steps have also resulted in analogy method provides good results and may be most suit-
new problems for the engineer, contractor, and supervisor. able for analyzing bridges with significant curvature. The fol-
This paper shows the different approaches on several factors. lowing paragraphs discuss some of the papers and reports
The paper is divided into three parts as follows: that were reviewed.
1. The design of bridges in classes for span and type with ref- Al-Rifaie, W. N., and Evans, H. R. (1979) An Approximate
erence to the pertinent factor for that design; Method for the Analysis of Box-girder Bridges that are
2. The contractor's approach to construction that illustrates Curved in Plan, Proc., Int. Association of Bridges and Struc-
the need for flexibility in the construction method in order tural Engineering, Int. Association for Bridge and Structural
to meet contract deadlines; and Engineering (IABSE), pp. 121.
3. The views of the supervising engineer and his means of
achieving a balance between the designer's intentions and An approximate method for analyzing curved box-girder
the contractor's proposals. bridges using the nodal section method is described. This

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method, originally developed for straight box-girders, has solutions by other methods. This method is most applicable
been adapted for curved box-girders and is useful for pre- to steel box-girders and is of little use to our project.
liminary design of these structures. The method developed
applies to simple-span, single-cell box-girders. Buragohain D. N., and Agrawal, B. L. (1973) "Analysis of
In this procedure, the transverse nodal section is idealized Curved Box-Girder Bridges," Journal of the Structural Divi-
as a plane frame. Nodes on the frame are assumed to be fixed sion, Vol. 99, No. 5, pp. 799819.
against translation but free to rotate. Each frame is analyzed
and the reactions at the nodes are determined. The reactions A discrete strip energy method is presented for the analy-
are then applied to longitudinal plates that represent the sis of curved box-girder bridges of arbitrary cross section and
components of the box-girder. The plates can only resist in- various forms of curved folded plate structures simply sup-
plane forces. The final step is to apply a "sway correction" ported at the two ends and composed of elements that may,
procedure that will make the displacements of the nodes in in general, be segments of conical frustra. The method de-
each of the transverse nodal sections compatible with the scribed applies to orthotropic material properties, arbitrary
deflections of the nodes along the edges of the longitudinal cross sections, constant curvature, and pinned supports at
plates. This approach results in substantial computational both ends. The method is based on harmonic analysis in the
savings over the finite element method and is well suited to circumferential direction. The total potential energy of the
preliminary design studies. structure is discretized into energy due to extension and
The method was checked against finite element results bending and energy due to shear and twisting. The two types
for model bridges representing both concrete and steel of circumferential strip elements are obtained by using a
box-girders. Several different span-to-radius ratios and modified finite difference discretization in the transverse
loading conditions were considered. In general, good cor- direction. The use of minimum energy principles yields two
relation was found for critical stresses, although some dis- types of element matrices assembled to form the overall stiff-
crepancy was found for non-critical stresses. The method ness matrix of the structure following stiffness matrix proce-
as developed does not accurately account for shear lag in dures. Results of two examples obtained by the method are
the deck. compared with available solutions. The applicability of this
Studies have extended this method to multi-cell box-girders paper to NCHRP 12-71 is limited because it only applies to
and have developed methods to account for shear lag in simply supported bridges, and the tool (software) to imple-
straight box-girders. These refinements are also being con- ment this method is not readily available.
sidered for curved box-girders.
Choudhury, D., and Scordelis, A. C. (1988) "Structural
Bazant, Z., and El Nimeiri, M. E. (1974) "Stiffness Method Analysis and Response of Curved Prestressed Concrete
for Curved Box-girders at Initial Stress," Journal of the Box-Girder Bridges," Transportation Research Record 1180,
Structural Division, Vol. 100, No. 10, pp. 20712090. Transportation Research Board, National Research Council,
Washington, D.C., pp. 7286.
A sophisticated numerical method of analysis of the global
behavior of long curved or straight single-cell girders with or A numerical finite element analysis method for linear-
without initial stress is presented. It is based on thin-wall elastic analysis and nonlinear material analysis of curved pre-
beam elements that include the modes of longitudinal warping stressed concrete box-girder bridges is demonstrated through
and of transverse distortion of the cross section. Deforma- two examples. A curved nonprismatic thin-walled box-beam
tions due to shear forces and transverse bi-moment are in- element is used to model the bridges. The cross section
cluded, and it is found that the well-known spurious shear of the element is a rectangular single-cell box with side
stiffness in very slender beams is eliminated because the inter- cantilevers. Eight displacement degrees of freedom, includ-
polation polynomials for transverse displacements and for ing transverse distortion and longitudinal warping of the
longitudinal displacements (due to rotations and warping) cross section, are considered at each of the three element
are linear and quadratic, respectively, and an interior mode is nodes. Prestressing, consisting of post-tensioned bonded
used. The element is treated as a mapped image of one parent tendons in the longitudinal direction, is considered. For
unit element and the stiffness matrix is integrated in three nonlinear material analysis, the uniaxial stress-strain curves
dimensions, which is numerical in general, but could be of concrete, reinforcing steel, and prestressing steel are
carried out explicitly in special cases. Numerical examples of modeled. The shear and the transverse flexural responses of
deformation of horizontally curved bridge girders, and of the box-beam cross section are modeled using trilinear con-
lateral buckling of box arches, as well as straight girders, stitutive relationships based on cracking, yielding, and ulti-
validate the formulation and indicate good agreement with mate stages. The first example demonstrates the versatility

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of the numerical method in determining the linear-elastic Computers and Structures, Inc. (1998) "SAP2000 Inte-
distribution of forces in a three-span prestressed box-girder grated Finite Element Analysis and Design of Structures,"
bridge of curved plan geometry and variable cross section. CSI, Berkeley, California.
Dead load, live load, and prestressing load cases are analyzed.
In the second example, overload behavior and ultimate This reference constitutes the concrete structure portion
strength of a three-span curved prestressed concrete box- of the SAP2000 Manual, with emphasis on design code check
girder bridge under increasing vehicular load are investi- analysis. SAP2000 features integrated modules for design
gated. The different response characteristics of the bridge of both steel and reinforced concrete structures. The pro-
induced by different transverse locations of the overload gram provides the user with options to create, modify,
vehicle are presented. analyze, and design structural models. The program is struc-
Although the finite element formulation might be detailed tured to support various design codes for the automated
and comprehensive and conducive to studying the ultimate design and check of concrete frame members. The program
behavior of concrete box-girder bridges, its applicability to currently supports several foreign and domestic design
the planned global elastic analysis studies is limited due to its codes. Given that the design code check features of the pro-
complex nature. gram focus on frame analysis, these design code checks are
of limited usefulness for the specialized needs of curved con-
Chu, K. H. and Pinjarkar, S. G. (1971) "Analysis of Hori- crete box-girder bridge "local" or sectional ("regional")
zontally Curved Box-Girder Bridges," Journal of the Struc- analysis. But the program is, of course, very useful for global
tural Division, Vol. 97, No. 10, pp. 24812501. analysis.
Chapter II of this reference outlines various aspects of the
A finite element method for the analysis of simply concrete design procedures of the SAP2000 program. This
supported curved girder bridges with horizontal sector chapter describes the common terminology of concrete
plates and vertical cylindrical shell elements is outlined. design as implemented in SAP2000. Each of six subsequent
Stiffness coefficients of sector plates are presented herein chapters gives a detailed description of a specific code of prac-
whereas stiffness coefficients of shell elements are based on tice as interpreted by and implemented in SAP2000. Each
Hoff's solution of Donnell's equations. The authors claim chapter describes the design loading combination, column
that this analysis is much more accurate than other methods and beam design procedures, and other special consideration
of analysis. required by the code.
Results of a sample bridge analysis are shown with stresses Aside from the obvious use as a SAP user reference, this
and deflections reported for a simply supported multi-cell document is useful as a summary (and side-by-side compari-
concrete bridge. Some interesting results, particularly those son) of various design codes for concrete columns and beams.
with respect to the effect of radius of curvature, were ob- Other than this, it is of limited direct utility to NCHRP Proj-
tained. Although a comparison is made of the results of a ect 12-71. There is no coverage of design or analysis of
curved twin box-girder bridge obtained by the proposed prestressing in this document.
method and another approximate analytical method (Tung,
1967), no comparisons with other (simpler) analysis methods Fu, C. C., and Tang, Y. (2001) "Torsional Analysis for Pre-
are given. The FEM analysis tool itself is not readily available stressed Concrete Multiple Cell Box," ASCE Journal of
and thus is of limited use, but the results of the analysis can Engineering Mechanics, Vol. 127, No. 1, pp. 4551.
serve as a comparison case for measuring the accuracy of
other methods, if more detail on the presented example can Using the Softened Truss Model, the authors present the
be obtained. formulation for calculating torsional effects in a multi-cell
reinforced and prestressed concrete box-girder bridge. This
Bridge Design System (BDS) (1986) A Computer Pro- paper asserts that because concrete box-girder sections are
gram for Analysis and Design of Multi-Cell Box-girder not made of thin webs and flanges, the stress distribution
Bridges, ECC. in these components is not constant and varies through the
thickness, causing the effective stiffness of the member to
The described software program is the most commonly used be less than that observed at low values of load (torque).
software for design of multi-cell box-girder bridges. Bridges are The formulation is coded in a computer program and the
modeled as plane frames ignoring all horizontal curve effects. results from an example problem are presented. This re-
This modeling technique is significant for NCHRP Project 12-71 search may be of some value to NCHRP Project 12-71 if the
because the technique is commonly used in practice and its methodology can be simplified and used as the basis of sim-
limits of applicability need to be investigated. plified methods for calculating torsional effects. However,

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the presented paper, in its current form, is too complex Meyer, C. (1970) "Analysis and Design of Curved Box-Girder
to be used in practical design situations or for parametric Bridges," Structural Engineering and Structural Mechanics
studies. Report No. UC SESM 70-22, University of California,
Berkeley.
Lopez, A., and Aparico, A. C. (1989) "Nonlinear Behavior of
Curved Prestressed Box-Girder Bridges, IABSE Periodica, The history of curved bridges and the highway geometric
Zurich, Vol. 132, No. 1, pp. 1328. requirements of these structures are discussed. The report
outlines the methods developed over the years for analyzing
This paper describes an analytical study of the ultimate curved bridges. These include straight beam approximation,
strength of horizontally curved reinforced and prestressed curved beam theory, refined curved beam theories, plate and
concrete box-girder bridges. The analysis was performed grillage analysis methods, finite element analysis, and the fi-
using materially nonlinear plane stress finite elements (i.e., nite strip method analysis of curved folded plates. Refined
panels) that exhibited membrane action. The material was curved beam theories are required to analyze thin-walled box
assumed to have a variable modulus of elasticity that was sections that can experience warping of the cross section in
strain dependent. Panel behavior was based on the evolutive the transverse direction. Because concrete box sections have
truss analogy with peak stress reduction (Vecchio and Collins, relatively thick walls, warping is generally small and ordinary
1986). Reinforcing steel and prestress strand were stressed curved beam theory can be used successfully.
uniaxially according to an assumed multi-linear stress Two methods of analysis are developed in the form of
strain relationship. Section warping was not considered. computer programs. The first program, FINPLA2, uses the
Classical matrix analysis techniques were used to perform the finite element method. The second program, CURSTR, uses
analysis. the finite strip method of curved folded plates. The solution
A five-span bridge was selected to study the difference be- methodology requires that loadings be applied in the form of
tween linear and nonlinear response. Live loads were located Fourier series. The programs yield essentially the same
at various transverse positions and the behavior was observed results.
as the intensity of these loads was increased. Based on these The CURSTR program was used to study wheel load dis-
studies the following conclusions were drawn: tribution in 1-, 2-, 3-, and 4-cell concrete box-girder bridges.
Several parameter studies were conducted with different cur-
1. The structural response was highly nonlinear at ultimate vatures, span lengths, deck widths, depth-to-span ratios, and
loads. loading configurations.
2. The form and degree of internal force redistribution at With respect to single-cell boxes, the following observa-
ultimate loads depended on the loading case considered. tions were made:
3. Internal forces were redistributed due to progressive
cracking and structural coupling between bending and 1. The girder on the inside of the curve is stiffer than the girder
torsion. on the outside of the curve and will attract more load.
4. The type of failure depended on the loading case considered. 2. Load distribution improves with an increase in curvature.
5. Ultimate internal force response could be evaluated accu- This behavior is independent of span, cell width, and
rately using plastic sectional analysis. depth-to-span ratio.
6. Transverse prestressing significantly affected post-cracking 3. The girder on the outside of the curve has a larger statical
response. moment because of its longer span.
4. The combination of items 1 and 3 results in nearly equal
The following criteria are proposed for design of curved moments in the two girders.
prestressed box-girder bridges. 5. The influence of span length on load distribution is simi-
lar to straight girders.
1. The response of the bridge under service loads can be 6. The influence of depth-to-span ratio is also similar.
accurately predicted using elastic models.
2. Elastic models cannot accurately (and will often non- For two-cell boxes:
conservatively) predict the ultimate limit state.
3. When using linear analysis to determine the factor of 1. The moments in the middle girder and the girder on the
safety against failure, cracked flexural and torsional sec- inside of the curve increase with curvature.
tion properties should be used to determine demands and 2. The moment in the girder on the outside of the curve de-
plastic sectional analysis should be used to determine creases with curvature up to a certain level and then starts
capacities. to increase.

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3. The influence of span on load distribution is small. Conclusions made in the paper are as follows:
4. Cell width accelerates the curvature effects.
1. A series of stiffness parameter equations and limiting angle
The response of 3- and 4-cell box-girders exhibits similar equations have been presented, which provide informa-
characteristics to 1- and 2-cell boxes. tion to the designer in determining the need for a curved
With respect to negative moments over a "fixed" support, girder analysis. The expressions are functions of the girder
types, bending and torsional stiffness, and central angles.
1. The girders may be assigned moments proportional to 2. The evaluation of gives the following criteria: "when is
their moments of inertia. less than 0.4, evaluation of stresses due to pure torsion
2. Load distribution is generally worse in continuous bridges. may be omitted. When is greater or equal to 10, evalua-
tion of stresses due to warping may be omitted."
For design, approximate methods are justified and even
preferred in most cases. A girder moment distribution factor Sennah, K. M., and Kennedy, J. B. (2002) "Literature Review
is developed: in Analysis of Curved Box-Girder Bridges," Journal of
Bridge Engineering, ASCE, Vol. 7, No. 2, pp. 134143.
B L
= 1+ 2.1 -
R 600 The curvilinear nature of box-girder bridges, along with
their complex deformation patterns and stress fields, have led
Bridges with curvatures radii large than 1000 ft. may be designers to adopt approximate and conservative methods
considered straight for analysis purposes. for their analyses and design. Recent literature on straight and
curved box-girder bridges has dealt with analytical formula-
Nakai, H., and Heins, C. P. (1977) "Analysis Criteria for tions to better understand the behavior of these complex
Curved Bridges," Journal of the Structural Division, ASCE, structural systems. Few authors have undertaken experi-
Vol. 103, No. 7, pp. 14191427. mental studies to investigate the accuracy of existing methods.
This paper presents highlights of references pertaining to
The paper reports on a series of stiffness equations and lim- straight and curved box-girder bridges in the form of single-
iting angle equations developed for determining the need for cell, multiple-spine, and multi-cell cross sections. The litera-
analysis of a bridge as a curved structure. The equations con- ture survey presented herein deals with (1) elastic analysis,
sider the type of supporting element, (i.e., open girder, spread and (2) experimental studies on the elastic response of
box, or single-cell box), bending and torsional stiffness and box-girder bridges.
central angles, and the induced stresses and deformation. It The elastic analysis techniques discussed include
appears that these equations are specific to steel girders and
warping torsion is a part of this methodology. However, the 1. Orthotropic Plate Theory Method
overall approach may be used or modified to apply to concrete 2. Grillage Analogy Method
bridges and NCHRP Project 12-71, especially for flowchart- 3. Folded Plate Method
ing the decision path for analysis. 4. Finite Strip Method
The paper provides equations for moment, stress, and de- 5. Finite Element Method
flection of curved and straight bridges. Design criteria for
curved bridges have been formulated using these equations, The orthotropic plate method lumps the stiffness of the
along with parametric studies. The range for the param- deck, webs, soffit, and diaphragms into an equivalent or-
eter , which "relates the cross-sectional geometry and thotropic plate. The Canadian Highway Bridge Design Code
the spacing between the outside girders, is determined for (Canadian Standards Association, 1988) recommends limit-
multiple I, twin box, and monobox-girders. Data for multi- ing this method to straight bridges with multi-spine cross
cell girders are not available from this paper. The bounds for sections. Parameter studies indicated that acceptable results
the torsional stiffness parameter are derived and depend are given for bridges with three or more spines.
on "the central angle," which is the total horizontal angle In the grillage analogy method, the multi-cellular structure
the girder passes through between supports, the torsional is idealized as a grillage of beams. The CHBDC does not rec-
rigidity of the cross section, and EI. The deflection ratio is ommend this method be used for sections with less than three
primarily dependent on , which "reflects the bending and cells or box beams. This method requires special attention to
torsional stiffness of the girders." The relationships between the modeling of shear lag and the torsional stiffness of closed
and the central angle were also found for the three bridge cells. When modeling is properly done, this method yields
types studied. results that compare well with finite element techniques.

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The folded plate method uses plates to represent the deck, of diaphragms, and loading was performed. The effects of
webs, and soffit of box-girders. Diaphragms are not modeled. these parameters on longitudinal stresses are considered,
The plates are connected along their longitudinal edges and based on selected numerical results. Implications for prelim-
loads are applied as harmonic load functions. The method is inary design are presented for both concrete and composite
time consuming and only applicable to restrictive support concrete/steel sections
conditions.
The finite strip method has been widely researched. It is Reilly, R. J. (1972) "Stiffness Analysis of Grids Including
essentially a special case of the finite element method but Warping," Journal of the Structural Division, ASCE, Vol. 98,
requires considerably less computational effort because a No. 7, pp. 15111523.
limited number of finite strips connected along their length
are used. Its drawback is that it is limited to simply supported Two methods of including warping effects in the stiffness
bridges with line supports and thus not applicable as a gen- method of analysis are presented. Method B seems to be
eral use analysis tool for production design. superior to Method A for cases where the warping constant is
With the advent of powerful personal computers and com- not large. In the limiting case where Iw = 0, the warping effects
puter programs, the finite element method has become the disappear and leave only the familiar GIx/L. When Iw is small
method of choice for complex structural problems. Many relative to Ix (approximately pL > 5 for each element) com-
researchers have applied this technique to the analysis of putational errors grow, because the stiffness matrix tends
curved box-girder bridges. A problem that occurs is that a to become singular as the elements on the main diagonal
large number of flat plate elements are required to properly associated with warping approach zero. This is not a serious
model the curved elements of a curved bridge. Several practical problem, as good solutions can be obtained using an
researchers have attempted to overcome this difficulty by ordinary grillage analysis, neglecting warping for structures
developing special elements or using special substructuring where warping stiffness is small and p is large. Composite
techniques. The versatility of this method has allowed re- bridges seem to fall near the borderline where warping can be
searchers to investigate several aspects of bridge behavior, in- neglected. The bridge used in the example above was non-
cluding dynamics, creep, shrinkage, and temperature. composite so that warping would be significant. Bimoment
Curved box-girder structures cannot be accurately analyzed and warping torsion are obtained for grillage structures. Re-
using the classical curved beam theory developed by Saint- sults of computer programs based on these methods are
Venant because it does not account for warping, distortion, shown to agree closely with published solutions for straight
and bending deformations of the individual wall elements of beams, a curved beam, and a curved highway bridge.
the box. Vlasov first developed an adaptation of Saint Venant
theory to thin-walled sections. Even this adaptation does not Meyer, C., and Scordelis, A. C. (1971) "Analysis of Curved
account for all warping and bending stresses. Considerable Folded Plate Structures," Journal of the Structural Division,
research effort has been expended over the years to develop Vol. 97, No. 10, pp. 24592480.
computational techniques to overcome shortcomings in the
present theory. A finite strip method of analysis is presented which can be
Several laboratory experiments involving model box-girder used to analyze curved folded plate structures simply sup-
bridges have been conducted over the years. In general, these ported at the two ends and composed of elements that may, in
experiments have shown good agreement with analytical general, be segments of conical frustra. The method is based
results, particularly those obtained using the finite element on a harmonic analysis in the circumferential direction, with
method of analysis. the loadings expressed by Fourier series, and on a finite ele-
In conclusion, the finite element method, though more dif- ment stiffness analysis in the transverse direction. The direct
ficult to apply, accounts for all relevant behavior in curved box- stiffness method is used to assemble the structure stiffness
girder bridges and yields the most reliable analysis results. Many matrix and to determine displacements and element stresses.
computer programs have been developed specifically for box- A description of a general computer program developed for
girder bridges, but most of these are not commercially available. the analysis and the results of several examples are also given.
Turkstra, C. J., and Fam, A. R. M. (1978) "Behavior Study of Okeil, A. M., and El-Tawil, S. (2004) "Warping Stresses in
Curved Box Bridges," Journal of the Structural Division, Curved Box-Girder Bridges: Case Studies," Journal of Bridge
ASCE, Vol. 104, No. 3, pp. 453462. Engineering, Vol. 9, No. 5, ASCE.
A numerical analysis of several single-cell curved box-girder This paper discusses case studies performed on 18 actual
sections with variable curvature, length, web spacing, number composite steel-concrete box-girder bridges. These analytical